So the other day Julia Galef and I had the pleasure of interviewing mathematical cosmologist Max Tegmark for the Rationally Speaking podcast. The episode will come out in late January, close to the release of Max’s book, presenting his Mathematical Universe Hypothesis (MUH). We had a lively and interesting conversation, but in the end, I’m not convinced (and I doubt Julia was either).

The basic idea is that the ultimate structure of reality is, well, a mathematical one. Please understand this well, because it is the crux of the discussion: Tegmark isn’t saying anything as mundane as that the world is best described by mathematics; he is saying that the ultimate nature of reality is mathematics.

This is actually not at all a new thesis, though Max is advancing it in new form and based on different reasoning then before. Indeed, the idea has a long philosophical history, and can fruitfully be thought of as based on two distinct philosophical positions: Pythagoreanism, or mathematical Platonism; and Mathematical monism.

Mathematical Platonism is the idea that mathematical structures are real in a mind-independent fashion. They are not “real” in the same sense as, say, chairs and electrons, but they do have an ontological status independent of the human (or any other) mind. As readers of this blog know, I’m actually sympathetic to (though not necessarily completely on board with) mathematical Platonism. The best point in its favor is the so-called “no miracles” argument, the idea that mathematics is too unreasonably effective (at predicting things about the world) for it to be just a human invention, rather than somehow part of the inherent fabric of the world. (Interestingly, this argument is equivalent to one by the same name advanced by scientific realists to claim that science really does describe — approximately — how the world is, as opposed to the antirealist position that the only thing we can say about science is that it is empirically adequate.)

Mathematical monism is the stronger doctrine that not only are mathematical structures real, but they are the only real thing out there (or, more precisely, everywhere).

The combination of Platonism and monism yields a class of theories about the ultimate nature of reality, of which Tegmark’s MUH is one example. We have seen another one several times in the past, in the form of Ladyman and Ross’ onticstructuralrealism, the notion that there are no “objects” or “things” at the bottom, just (mathematical) relations.

While I have commented positively on ontic structural realism (again, without necessarily buying into it), and more generally on the idea of a “naturalistic” metaphysics (i.e., a metaphysics that takes seriously the best known physics), my conversation with Max Tegmark actually generated more doubts than illumination.

One obvious problem is posed by what it would mean for the world to be “made of” mathematical structures. The notion of mathematical structure is well developed, so that’s not the issue. A structure, strictly speaking, is a property or a group of mathematical objects that attach themselves to a given set. For instance, the set of real numbers has a number of structures, including an order (with any given number being either less or more than another number), a metric (measuring the distance between points in the set), an algebraic structure (the operations of addition and multiplication), and so on.

The problem is in what sense, if any, can a mathematical structure, so defined, actually be the fundamental constituent of the physical world, i.e. being the substance of which chairs, electrons, and so on, are made.

Of course, both Julia and I asked Max that very question, and we were both very unconvinced by his answer. When Tegmark said that fundamental particles, like electrons, are, ultimately mathematical in nature, Julia suggested that perhaps what he meant was that their properties are described by mathematical quantities. But Max was adamant, mentioning, for instance, the spin (which in the case of the electron has magnitude 1/2). Now, the spin of a particle, although normally described as its angular momentum, is an exquisitely quantum mechanical property (i.e., with no counterpart in classical mechanics), and it is highly misleading to think of it as anything like the angular momentum of a macroscopic object. Nevertheless, Julia and I insisted, it is a physical property described by a mathematical quantity, the latter is not the same as the former.

Could it be that theories like MUH are actually based on a category mistake? Obviously, I’m not suggesting that people like Tegmark make the elementary mistake of confusing the normal meaning of words like “objects” and “properties,” or of “physical” and “mathematical.” But perhaps they are making precisely that mistake in a metaphysical sense?

There are other problems with MUH. For one, several critics of Tegmark’s ideas have pointed out that they run afoul of the seemingly omnipresent (and much misunderstood) Gödel’s incompleteness theorems. Mark Alford, specifically, during a debate with Tegmark and Piet Hut has suggested that the idea that mathematics is “out there” is incompatible with the idea that it consists of formal systems. To which Tegmark replied that perhaps only Gödel-complete mathematical structures have physical existence (something referred to as the Computable Universe Hypothesis, CUH).

This, apparently, results in serious problems for Max’s theory, since it excludes much of the landscape of mathematical structures, not to mention that pretty much every successful physical theory so far would violate CUH. Oops.

Prompted by the above, I also asked Max about Gödel, and his response was that Gödel-related problems appear only in the case of infinite quantities, and he professed himself to be an infinity-skeptic. That took me by surprise, what do you mean you don’t believe in infinity? I thought this was a pretty darn well established concept in mathematics, at least since the work of Georg Cantor in the 19th century! But of course Tegmark was referring to the existence of physical, not mathematical, infinities. As is well known, there are certain calculations in physics that do generate infinities, for instance the singularity that shows up in the description of black holes, or the infinite quantities that are postulated in standard descriptions of phase transitions. The question of whether there really are infinities in physical systems is open, so surely Max is entitled to his skepticism. But it did seem a bit too convenient a position, in light of the above mentioned Gödel-related problems.

Another issue that didn’t convince either Julia or me during our conversation with Max is a crucial one: testability. I’m okay with philosophical speculations (and I use the term in a positive fashion!) about modalrealism or the principle of plenitude, but if we are claiming to be doing science (as Tegmark surely is), then our speculations better make contact with empirical reality. Jim Baggott, in his Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth, is already accusing physicists of losing touch with what it means to do science. Is Tegmark the latest example of the trend?

When we asked, he claimed that the MUH does make empirical predictions, but when pressed on the details the answer becomes far less satisfying than one would hope. For instance, Max said that one prediction is that physics will continue to uncover mathematical regularities in nature. Well, probably, but one surely doesn’t need to postulate MUH to account for that. He also has stated in the past that — assuming we live in an average universe (within the multiverse of mathematical structures) — then we “start testing multiverse predictions by assessing how typical our universe is.” But how would we carry out such tests, if we have no access to the other parts of the multiverse?

Max went on to say that his hypothesis has “zero free parameters” and is therefore favored by Occam’s razor. But if you check his paper at arxiv.org he says: “If this theory is correct, then since it has no free parameters, all properties of all parallel universes … could in principle be derived by an infinitely intelligent mathematician. … Finally, the ultimate ensemble of the Level IV multiverse would require 0 bits to specify, since it has no free parameters.” There are a couple of obvious problems here. One is the dearth of infinitely intelligent mathematicians, the second the fact that the above mentioned Level IV multiverse is precisely what gets dramatically (and unrealistically) shrunk as a result of Gödel-imposed limitations. And let’s not forget that Occam’s razor is just a useful heuristic, it should never be used as the final arbiter to decide which theory is to be favored, especially when we are talking about such highly speculative and empirically next to impossible (or even downright impossible) ideas to test.

In Many Worlds in One: The Search for Other Universes, critic Alex Vilenkin says that “the number of mathematical structures [in the multiverse] increases with increasing complexity, suggesting that ‘typical’ structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world.” In order to get around that problem, Tegmark assigns lower weights to more complex structures, but since this is done without a priori justification, it is an ad hoc move, which of course violates Occam’s razor. So, as much as I enjoyed our conversation with Max, for the time being I remain skeptical of the MUH and related hypotheses. Maybe we just need to wait for the appearance of an infinitely intelligent mathematician.

_______[This just in from Max Tegmark himself!]

Thanks Massimo for the fun conversation during the interview and for raising these important questions! They are excellent ones, and a key reason why I spent three years writing this book is because I wanted to make sure to finally answer them all properly. Needless to say, I couldn't do justice to them in our short interview, so I'm very much look forward to hear what you think about my detailed answers in chapters 6, 10, 11 and 12. I think you'll find that our viewpoints are closer than your post suggest - for example, your statement "Tegmark assigns lower weights to more complex structures" is not something you'll find in the book. Rather, I describe how the measure problem is a terrible embarrassment for modern cosmology (regardless of whether the MUH is true or not) that we need to solve, and that our untested assumption that truly infinite things exist in nature are my prime suspect: we've never measured anything to better than 17 decimal places, have only 10^89 particles in our universe, and manage to do all our publishable physics simulations with computers that have finite resources, so even though my physics courses at MIT use infinity as a convenient tool, I respectfully object to your "OPS" argument that we somehow have experimental evidence for infinity in physics. Without infinity, there are, as you say, no Gödel issues in our physics.

Comments

I'm with Max Tegmark on the infinities, but I think he's making a big mistake pursuing the mathematical universe. His paper on arXiv dates from 2007, and we can all cut a guy some slack for coming up with a wild unscientific idea that gets him some attention. But to be still pushing it seven years later doesn't make him look so good. There's all sorts of words I could use at this juncture, but I won't.
By the way, it isn't misleading to think of electron spin like the angular momentum of a macroscopic object. Remember the electron magnetic moment, check out the Einstein-de Haas effect, and then take a look at Dirac's belt and the spindle-sphere torus.

Thanks John Duffield for joining in. You’re certainly correct that electron spin differers from macroscopic angular momentum, but what physicist are you criticizing for claiming that there’s no difference? Can you provide a link? Also, I’ll be happy to address the criticisms you have of the ideas I explore in my book if you’re willing to tell me specifically what they are.
/Max

Hi Max. I wasn't criticizing anybody in particular about spin, and I was pointing to the evidence that suggests that electron spin isn't that different to macroscopic angular momomentum. For example the Einstein-de Haas effect "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies". We don't hear much about that, instead we get the non-sequitur as per this old Stern-Gerlach article: "If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast". It isn't rotating the way a planet rotates!

I haven't read your book I'm afraid. I read your paper some time back. I'm afraid I wasn't keen on it. You mentioned the fine structure constant which is a running constant and so isn't a constant. You referred to parallel universes which are beyond the reach of science. And you didn't say matter is made of energy or wavefunction or light or motion, or that you cannot distinguish space and energy at the fundamental level. You talked instead of mathematical structures. The electron is arguably a 4πn/c^1½ standing-wave harmonic quantum field structure where rotation at c and orthogonal rotation at ½c sweeps a 4π solid angle, but it isn't made of mathematics. I'm sorry Max, I think you're onto a hiding for nothing with that kind of thing, and that you shouldn't let it distract you from some really interesting stuff going begging. Remember a few years back on Horizon when you referred to the waterfall analogy? IMHO you should examine that some more, and Friedwardt Winterberg's firewall. And the AMPS firewall and the information "paradox" and "Hawking radiation" and the cosmological "constant". And conservation of energy re dark energy, and inhomogeneous vacuum energy and mass equivalence, and the edge of the universe. The list goes on. If you've covered some of this in your book that's good, but Our Mathematical Universe puts me off I'm afraid. Sorry.

You write: "we can all cut a guy some slack for coming up with a wild unscientific idea ... But to be still pushing it seven years later doesn't make him look so good."

I find it odd you say this, and then immediately start spouting wildly unscientific ideas. I've also noticed that whenever someone pushes you to actually show some math to back up your claims, you avoid or go silent. You claim you understand modern physics and can do the math to calculate predictions from these theories, but it has become very clear to people here that your knowledge goes no deeper than the convoluted collection of layman's descriptions and anologies that you keep spouting.

So either stop using science20 comments as your platform for your nonsense, or start responding to requests to actually back up your claims. I think Edward picked a perfect one for you to start with, as your claim that the invariant mass of a point particle falling in GR is not invariant is so simply checked and seen to be incorrect, that it boggles the mind that you continue on with your beliefs without actually working out the predictions yourself to check them.

"demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies"

Stop arguing by quotes. You treat science as if it is liturgical exercise arguing by scripture. Do you even know what that quote means? By "same nature" it does NOT mean that spin is literally a rotating body, the point is that spin and a rotating body both carry angular momentum, and that it is the combination of these angular momentums, the total angular momentum, that is conserved. This was a big clarification of understanding at the time. That is the way in which they are the "same nature". They are distinct however in that spin is an instrinsic angular momentum. Apply any torque you want, and the magnitude of the spin angular momentum of, say, an electron will not change. This is very unlike normal classical moving bodies angular momentum. So it is incorrect to view spin as some kind of classical angular momentum, and even more bizarre and incorrect to view an electron as a photon twisting around. To have a photon self-interact like that would require a crazy new term in the electrodynamics Lagrangian. Your ideas are all over the place and wrong, so please stop pushing your pet theory talking points on this science site.

I didn't start "spouting wildly unscientific ideas". What I said was this:
"By the way, it isn't misleading to think of electron spin like the angular momentum of a macroscopic object. Remember the electron magnetic moment, check out the Einstein-de Haas effect, and then take a look at Dirac's belt and the spindle-sphere torus".

As for photon interaction, see two-photon physics on wiki and note that the given explanation is that one of the photons spontaneously transforms into an electron-positron pair. That's like worms from mud, and it's wrong. Photons do not spend their time spontaneously morphing into electron-positron pairs which then magically transform back into a single photon which nevertheless manages to keep on propagating at c. Pair production does not occur because pair production occurs. That's crazy.

Now if you don't mind, this thread is about Max Tegmark, not me. So no more please.

Your ideas on substructure of the electron are indeed wildly unscientific. It is pure crackpot. You don't even understand enough physics to understand the objections people give, as is clear from your response to my comments on photon interactions. If you feel we are misunderstanding your ideas, then stop using vague logically disconnected statements, and specify your theory precisely in a mathematical form. Then no one can dispute the predictions of your theory, and hopefully even you could finally see that your ideas don't work. Show us a Lagrangian that can back up any of your claims.

> Now if you don't mind, this thread is about Max Tegmark, not me. So no more please.

YOU are the one that started pushing your crackpot ideas here. In a reply to Max you even elaborated on your electron is made of photons idea. I am requesting that you, instead of continually bringing up these ideas everywhere on science20, that you confine your crackpot non-sense to writing a blog about your pet theory of everything. Specify the theory precisely enough that there is no dispute calculating the predictions of the theory, and then learn from the failure of your theory and stop posting about it. Everyone wins.

If you are unable or unwilling to do that, then just please shut up about your pet crackpot theories, which despite your blindness, are indeed wildy unscientific ideas.

"...Your ideas on substructure of the electron are indeed wildly unscientific. It is pure crackpot..."

They aren't my ideas, they come from people like Williamson and van der Mark. Others have offered similar models, which can arguably be traced back to Thomson and Tait. And they aren't crackpot. We can create an electron along with a positron out of light in pair production. We can diffract electrons. In atomic orbitals electrons "exist as standing waves". When we annihilate the electron with the positron the result is light. So what do you think the electron is made of? Cheese? Magic? Mathematics?

And what's your name? I'm sure you're aware that you've come across as somebody whose physics knowledge is so scant that you can only offer feather-spitting anonymous ad-hominem abuse instead of counterargument and counterevidence. That cuts no ice I'm afraid. Not here or anywhere else.

I read the discussion, and I'm afraid Anon Poster is right. (Some people just don't want to reveal their identity online, whether their physics knowledge is scant or not; though I admit that using true name adds credibility, which why I'm doing this.)

So you think electron is made of photon, and your evidence is they can transform into each other, electron has the property of waves, etc.. But all of these have been explained successfully in the Standard Model without appealing to your (fairly simple-minded, sorry) idea. And here is one easy counterargument toward your "electron has the property of waves" evidence. Since you are well-read in physics, I believe you have at least heard of the term "wave-particle duality," which dates back to the 1900s; it says all matter has a wave-like nature, but it doesn't say that all wave-like stuff is light.

You cited some sources for the ideas you are promoting. I didn't read them though. First, physicists don't cite from cybsoc.org (what is that?) and scribd.com, or fiascopress.org, or Journal of Swarm Scholarship (again, what is that?)—if the papers appear there, chances are that they are crackpots. Second, sorry, I don't believe an electronic engineer has something valuable to say about TOE.

So my suggestion is if you want to learn modern physics, learn it the right way. Learn the Standard Model first if you really want to talk about the things you are talking. Your profile says, "maths relatively weak and technically an amateur," then I'm afraid you can't really understand anything about the Standard Model; do some homework first, maybe starting from http://theoreticalminimum.com/, which is designed exactly for people who have "life-long interests in physics" but never got a degree. (Prof. Lenny Susskind from Stanford is one of the fathers of String Theory; I heard his online courses—linked above—are great, though I never watched them myself. Anyway, Prof. Susskind taught me before, and I assure you that his lectures are thought-provoking for students of all levels of expertise). Ah, one more piece of suggestion: never, never, never, never, never try to learn modern physics from Wikipedia; and better yet, also abstain from popular science—they will lead to more wildly unscientific ideas.

At last, I have to admit that I'm not an expert. I'm an undergrad math and physics major, currently learning QM, QFT, GR, SM, etc. myself. But at least I'm on the right way. Experiences of myself and many mathematicians/physicists around me, including established ones, all suggest that there is no royal road to modern physics—you have to do the math. Basically you learn everything at least twice—the first time, you get lost in calculations; the second time, you begin to get some insight. So, the right way to the frontiers of physics is ahead of you; why not give it a try yourself.

All points noted Kevin. But with respect, the hard scientific evidence is there. We can make an electron along with a positron out of light, we can diffract it, it has its magnetic moment, the Einstein-de Haas effect is real, in atomic orbitals electrons exist as standing waves, and we can annihilate the electron with the positron to get light again. Doing the math doesn't make that evidence go away. Nor will doing the math explain gamma-gamma pair production. If you beg to differ, try giving your standard-model explanation. See above. The explanation you will give will say that one of the photons spontaneously transforms into an electron-positron pair, like worms from mud. As if photons are forever morphing into electron-positron pairs which then magically transform back into a single photon which nevertheless manages to propagate at c. You will assert that pair production occurs because pair production occurs. Your explanation will be tautological, and it will be wrong. And in the same breath you will dismiss those who offer another explanation as likely crackpots, you will not ask David St John about his physics degree, and you will say wildly unscientific ideas concerning an electron that was literally made from light without even commenting about a universe made of mathematics. Kevin, I would urge you examine the evidence and think for yourself instead of dismissing it in favour of "you have to do the math". Only then will you begin to get some insight. If you do not, like others before you, you will spend fruitless decades wasting your time whilst physics withers on the vine. By the way, the electron is not a point particle, it was Thomson and Tait who coined the phrase spherical harmonics, Ehrenberg and Siday wrote an electron-optics refraction paper that predicted what's now known as the Aharonov-Bohm effect, and science advances one funeral at a time.

Hi John, I won't argue about the theory/evidence any further, but I wish you could read these (no offense): Ten Signs a Claimed Mathematical Breakthrough is Wrong by Scott Aaronson, The Crackpot Index by John Baez, and The Alternative-Science Respectability Checklist by Sean Carroll. The moral is, even if you really have a ground-breaking result, mathematicians and physicists would be reluctant to even take a look at your result if the way you present it, and unfortunately, your degree, are not convincing enough (otherwise we'd have no time for research). So, if you really want your voice heard, you need to learn and respect the ways we do things. (I'm sorry, but isn't this the same in every community?) Note in particular that The Crackpot Index says,

10 points for each statement along the lines of "I'm not good at math, but my theory is conceptually right, so all I need is for someone to express it in terms of equations".

Moreover, you need to learn what we know before you could answer our questions and convince us about your theory. This is really hard, but if you want to do this, The Theoretical Minimum by Prof. Lenny Susskind—as I referenced in the post above—might be a good starting point. Prof. Gerard t'Hooft, a Nobel laureate, also compiled a list of topics that you should learn as a theoretical physicist, along with freely available references: HOW to BECOME a GOOD THEORETICAL PHYSICIST. I'm not sure if it's good or not, but I'm inclined to trust t'Hooft's taste.

Overall, your enthusiasm towards physics is very respectable. However, if you satisfy your enthusiasm in the hard yet fruitful way (uh, math), your time might be much better spent, and turn out to be much more enjoyable. And one day you might even be able to convince us. But before that day really comes, please do not make the mistake of claiming something along the line of "you will spend fruitless decades wasting your time whilst physics withers on the vine." Because it falls into the following category in the The Crackpot Index:

40 points for claiming that when your theory is finally appreciated, present-day science will be seen for the sham it truly is. (30 more points for fantasizing about show trials in which scientists who mocked your theories will be forced to recant.)

Again all points noted Kevin. I'm familiar with the above. Please note though that I'm not some "my theory" guy - I point out the scientific evidence and refer to the people who have done the work. Instead I'm a guy who's concerned about the state of physics, and about science in general. Sometimes I feel like we're heading into a dark age.

John, regardless of where your collection of ideas come from, can we agree that the scientific method suggests we should compare different hypotheses by comparing the predictions of those ideas to the results of experiments? We have learned quite a bit, enough to rule out entire "classes" of theories. The problem is that the ideas you are pushing are wrong, but you seem to be evaluating them at some level other than predictions compared to experiments. For example, I've seen you rant multiple times now on how pair production "must" mean electrons are made of photons. You are arguing from basically an 'a priori' expectation. That is not science, that is pseudo-science.

There are things that our current theories don't explain "well" and they are subject to intensive research, but the stuff you keep bringing up are already well understood (even if you don't understand the math). For example, your electron made of photons theory is nonsense. The electromagnetic interaction, as modeled by QED, has been incredibly well tested and passes muster so far. You instead are implying a photon can self-interact to wrap onto itself to make an "electron". I can speculate on all kinds of ways to try to model such an idea, but none of them would match experiment. What are the details of your model? You refuse to put anything down precisely though, and furthermore refuse to actually work out any math to back up your claims.

I think your claims about a point particle's invariant mass changing as it free falls to be a perfect example of the problem here. You, for whatever reason, are unwilling or incapable of doing the math to see if the theory actually predicts what you claim. No amount of explanation seems to be able to convince you that you are wrong. You are so steadfast in your belief of what the theory "must" predict, that you believe this despite all explanations given to you, and, most importantly, despite actually working out the math of the predictions yourself. Do you truly not see a problem in that? Do you not understand why this attitude keeps you stuck in incorrect understanding of physics?

Write a blog working out the calculations for your claim that the invariant mass of a point particle changes as it free falls according to GR. Actually work out the math. People can tell you the correct answer all you want, but until you actually see it yourself, it appears you will never actually learn your mistake.

Yes, we should employ the scientific method and predictions, but referring to pair production and electron diffraction & refraction and magnetic moment and atomic-orbital standing waves and Einstein-de Haas and annihilation is not “ranting”. It’s pointing out the hard scientific evidence. Dismissing this evidence is misguided. Believing that gamma-gamma pair production occurs because pair production occurs is misguided too, as is believing the electron is a point particle, as is your dismissal of the mass deficit. That isn’t “my idea”. You’ve already conceded that dropping a 1kg brick into a black hole will increase the black hole mass by only 1kg have you not? And you know that if you intercept the falling brick and radiate away its kinetic energy into space, then release the brick to complete its fall, the black hole mass increases by less than 1kg. You know the same is true for an electron, and that there is no mathematics that changes this. So you must know that attempting to hide behind a “show me the math” smokescreen will not rescue you here. By the by, your ad-hominem criticism of me and the evidence I refer to, and your lack of criticism of Max Tegmark’s universe made of maths does you no credit.

"You know the same is true for an electron, and that there is no mathematics that changes this."

Why are you so sure of the prediction before you actually work out the math? If you actually work out the math, it shows that your claims do not match what the theories actually predict. This is why everyone keeps referring to them as "your ideas". LFB already explained how your electron mass changing idea conflicted with precision atomic spectroscopy experiments. The scientific method shows that your ideas are wrong. What will it take for you to accept that?

"Yes, we should employ the scientific method and predictions"

Then work out the predictions and compare them to experiment. Making a series of logically disconnected statements and then claiming that people are "dismissing this evidence" is nonsense. None of the things you bring up support your claims. If someone posted a blog showing the math that a point particle that is free falling, then a force is applied, then allowed to freefall again, will still have the same invariant mass the whole way ... would that convince you? Why are you so certain that "no mathematics" can change your mind? What are you basing your predictions on then?

I haven't made a series of logically disconnected statements, I've pointed to the hard scientific evidence which you continue to dismiss, just as you continue to describe the electron as a point particle. It is not. Its field is what it is, it's quantum field theory, not quantum point-particle theory.

Why are you so sure of the prediction before you actually work out the math?

Conservation of energy. And I reiterate, the mass deficit is not "my idea". Note this:"However, binding energy lost from the system (as heat radiation) would itself have mass, and directly represents the 'mass deficit' of the cold, bound system". This is robust mainstream physics, and you know it. So your attempt to portray this as my idea and claim that the scientific method shows it to be wrong is dishonest, as are your anonymous ad-hominems. May I remind you that the falling brick is not subject to an external force. No energy is supplied to it. We say gravitational potential energy is converted into kinetic energy. This "gravitational potential energy" is not located in the black hole, or the intervening space, or anywhere mysterious. It's in the brick, and it has a mass-equivalence. As I've said previously system momentum p=mv is conserved, but KE=½mv² means we can discount what happens to the black hole and focus on the brick. When we dissipate its kinetic energy, its mass is reduced, just as a radiating body loses mass. LFB has explained nothing, and you do not understand that his "scientific method" is effectively weighing the brick using another brick as a counterweight.

> "I've pointed to the hard scientific evidence which you continue to dismiss"

Hard scientific evidence would be to calculate the predictions from your hypothesis, and compare it to experiment. Instead you just make statements and then your conclusion, without logic connecting them. That is not "hard scientific evidence", that is just you stating your conclusion and declaring it self-evident (to you). That is not how scientific claims are made. This is why working out the math is necessary, it forces precision, and you can no longer hide behind your vague descriptions. This is not an ad-hominem attack. It is an attack on your logic (or lack there-of in this case).

> " the mass deficit is not "my idea". "

Your belief that the invariant mass of a system decreasing means the invariant mass of the particles decreased ... that is NOT "robust mainstream physics". That is a misunderstanding of physics, and is very much "your idea".

> "This "gravitational potential energy" is not located in the black hole, or the intervening space, or anywhere mysterious. It's in the brick"

You can try to define such a thing, but in general in GR it requires a pseudo-tensor. Its not worth going into that, until you understand the basics of GR first. And even then, using a pseudo-tensor to describe "energy" due to the gravity field, the invariant mass of a free falling particle is still constant.

> Why are you so sure of the prediction before you actually work out the math?
> "Conservation of energy."

Conservation of energy in GR does NOT work like you apparently think it does.
Let's try a simple counter example, consider the free falling coordinate system in which the particle is at rest. It's kinetic energy is zero for the entire trip, and there is no force on it. It's invariant mass is constant. Do you still deny this? If so, actually work out the math for the prediction and post it in a blog here on science20.

Again, I ask: What could make you consider you are wrong if you are unwilling to actually calculate the predictions of GR?

If I recall correctly, doesn't Hilbert say "there are no infinities in nature" in his Infinite Hotel paper? If you somehow discovered a set of real things that you hypothesized to be infinite how would you verify it? You could count the objects forever and still have forever to go still unable to verify the set of objects was infinite.

Hi Hontas - in chapter 12, I discuss the Matrix and the famous simulation argument that you're referring to, and argue that there's a logical flaw in it. But if you're nonetheless really worried about being in a simulation right now, I suggest living as interesting a life as you can to minimize the probability that your simulator gets bored and shuts you down! /Max

The universe seems to be best described by quantum physics (QFT, quantum field theory). Its language is largely mathematical, but rigorous (exact) mathematics does not work in QFT. At best, maths suggests plausible ways how things work, but the results can't be proven to be correct.

In my opinion, the reason for this is that mathematics, with its base in idealized concepts, must ultimately be rooted in physical objects, such as the very paper you write on, or the language uttered to express results.

All such tangible things are governed by quantum physics, which generally does not allow precise borders between matter, space, time etc.

Therefore, it is physics that has the last say on the validity of anything deduced from mathematics, not vice versa. In that sense, mathematics is governed by the natural world, rather than the other way round.

Is there only one "mathematical structure", on which a world can be based? There must be many other possibilities just in the currently known mathematics. It reminds me of the 10^500 possibilities in the string theory, with the difference that this should be more than a theory.

It all sounds to me as just another anti-realist argument, complete with all of its ridiculous baggage. For example the anti-realist would argue that it is our language which drives reality, mathematics can be argued to be little more than a specialized language. In other words, our Earth was really indeed physically flat until Copernicus, and the language of his heliocentric model, physically changed it.

Political correctness has been the most obvious result of this philosophic perspective, if you can control the language then you can control reality, i.e. if you can eliminate racist language then you can eliminate racism. Similarly, if you can mathematically prove a theory, then the theory itself must be true. A HUGE mistake, just ask Aristotle.

Seriously, if you ever wish to be taken seriously then you should find a more reputable source than Wikipedia.

Perhaps I should have used a more general term like geocentric model promoters, of which the flat earther's are a subset of, mia culpa. By cherry picking only a partial quote (misquoting) from my comment are you trying to falsely argue by suggesting that this quote was my opinion? I can assure you that it is not, rather it is the anti-realist purists view. Unfortunately there are very large numbers of people who subscribe, at least partially, to this twisted philosophy, including many modern mathematicians and theorists.

Any idealist refutes or ignores basic truths and realities, lest they would become realists.

If you wish to research anti-realism I would suggest staying away from many of the obvious internet sources, especially Wikipedia. Since the anti-realists do not like this label they have co-opted many of the web sites and usually only offer strongly biassed opinions. Instead I would suggest only using reliable reputable sources of philosophy texts.

"if you ever wish to be taken seriously then you should find a more reputable source than Wikipedia."

I don't think you've read my column. I try, as best I can, to make science accessible to people from seven to centenarian. My projects are mostly built from construction toys, science toys, stuff just lying around the house or inexpensively acquired from closeout stores and second-hand stores. My articles are not intended as recipes since you may not have the same parts. I hope instead to inspire people to come up with their own designs so, wherever practical I try to publish my successes and my failures, so that whoever is interested can improve on both. I don't expect to be believed, since you can do it yourself. Wikipedia is just a starting point--the good stuff is in the footnotes.

"Perhaps I should have used a more general term like geocentric model promoters, of which the flat earther's are a subset of,"

Geocentric model would have been better. No educated person of the ancient world believed the earth was flat probably going as far back as Pythagoras.

The Copernican model was not widely accepted until Newton and there were still geocentric holdouts as late as 1838 until Bessel was finally able to measure stellar parallax putting the final nail in the coffin of the geocentric model. If you think about it, there were Steady State holdouts until the Cosmic Microwave Background radiation put the final nail in the coffin of Steady State theory.

"mia culpa." There's no reason for you to feel intense guilt for using "flat earthers" as an example. I've met a flat earther and he didn't seem sincere in his belief the earth was flat. It was more tongue-in-cheek. A reason to argue while drinking lots of beer.

I have no dog in the realist v. anti-realist fight. I leave that up to the folks in the humanities like philosophy to get it sorted out. If you have a citation for the "anti-realist purists" with regard to the geocentric model, I'll take a look at it.

Wow, absolving me of my sins from a self proclaimed priest, you just keep bolstering my general theories that most outspoken academics, and especially PhD's, think of themselves as pious religious leaders and are indeed merely practicing their religion and are NOT practicing science. Thank you for the generous extension of your sacrament.

What, if anything, have you actually contributed to this conversation, other than a demonstration of a narcissistic superego with a superiority (god) complex? (very common indeed)

"Wow, absolving me of my sins..." You don't seem to have a sense of humor.

I'll let other folks respond to this:

"you just keep bolstering my general theories that most outspoken academics, and especially PhD's, think of themselves as pious religious leaders and are indeed merely practicing their religion and are NOT practicing science."

Excuse me while I go pop some popcorn so I can sit back and watch the show.

Basically I agree with Tegmark’s idea, although I have not seen his book yet. The reason is that eventually all these so called “creations of mind” finally end up in something like cell phone or iPad which we can hold in hand and are as real as one can get!! So there must be some reality associated with the mathematical description. I have absolutely no sympathy with the saying that quantum mechanics is just a mathematical idea and one should not read reality in it. It is counterintuitive and totally inconsistent with our everyday life. But nobody can say that our intuition based on our experiences, with objects seen by us which are our size, should work for immensely small or immensely large objects.

I definitely agree with Kashyap and think that Tegmark is on to something here. Everyone who gets all up in arms about the un-physical nature of this theory doesn't seem to understand that modern theoretical physics has at its foundation point particles with many different properties that can only be abstracted away mathematically. Platonism is on very solid ground. I'll be reading this book

"Suppose that everything that could exist, does exist. The multiverse is not a bug, but a feature. We have to be careful: the set of everything that could exist belongs to the realm of metaphysics rather than of physics. Tegmark and I have shown that with a minor restriction, however, we can pull back from the metaphysical edge. Suppose that the physical universe contains all things that are locally finite, in the sense that any finite piece of the thing can be described by a finite amount of information. The set of locally finite things is mathematically well-defined: it consists of things whose behavior can be simulated on a computer (more specifically, on a quantum computer). Because they are locally finite, the universe that we observe and the various multiverses are all contained within this computational universe."

An interesting read, Phil. I think it's worth contrasting Seth Lloyd's Edge essay with Max Tegmark's. He suggested we should "retire" infinity. I rather liked it, and would like to see Max explore it further. For example, if space is not infinite, and then if space is "flat" as WMAP suggests, then space might have some kind of... edge. The irony is delicious.

I wrote a speculative piece about this on something called Bogpaper. I'm no longer writing for them, so I hope it isn't improper of me to mention it here now.

not to mention that pretty much every successful physical theory so far would violate CUH.

I'm not aware of any physical theory that provably relies on incomputable quantities, ie. that the computable real numbers don't suffice.

And let’s not forget that Occam’s razor is just a useful heuristic, it should never be used as the final arbiter to decide which theory is to be favored

This is incorrect. Occam's razor was formalized as a true arbiter of truth in Solomonoff Induction, which uses Kolmogorov complexity to totally order candidate theories, and provably converges on truth because of this property.

"My own view is that ultimately physical laws should find their most natural expression in terms of essentially combinatorial principles, that is to say , in terms of finite process such as counting.." Roger Penrose in Magic Without Magic. (and I agree with him ..tom)

Reality exists hence we say it is true. But what is really true besides that more than anything else which we can really trust, it is mathematical facts. So, to my mind I connect both since both seem to be a statement of truth. So I took a guess that reality is something akin to a circle (truth).

Wow. Someone actually quoted from
“Magic without Magic” in honor of John Wheeler's 60th
birthday. He speculated that mathematics is the eerie center of
physics … He is the coiner of “Black Holes” and “Farewell to
Geometry” (not to be mistaken with Farewell to Reality).

I have wondered if the mathematical
universe is a one-sided notion with the other side lying in the
second of C.P Snow's Two Cultures. The fiftieth anniversary of his
famous two-worlds two-cultures essay was in 2011, which
coincidentally was the fiftieth year for Stephan Wolfram, creator of all things
computable in Mathematica.

Let's ponder for a moment whether there can be things, even entire cultures that are NOT computable.

Against my will right now, I am
suffering through my annual accounting duties for IRS reports. After
30 years of doing it, I really am tired of the process. However, I
can understand why some people are drawn to the field of accounting.
The reason is that numbers don't lie. There is a purity is tallying
up expenses and income sources. There are grey areas for sure,
however, they originate from Snow's not-so-computable smokey worlds of unethical wheeling and dealing. They come from the un-mathematical
universe.

Mathematics is chock-full of straight
lines, perfect circles and infinities …. and yet none of these
things exist … so it can only be a tiny subset of mathematics that
is physically possible …. and even that set may be only a slice of
a totality, most of which is not computable.

The irony is that, as Nietzsche put it
(in Human, All Too Human), “Mathematics . . .
would certainly have not come into existence if one had known from
the beginning that there was in nature no exactly straight line, no
actual circle, no absolute magnitude.”

If I were to put
forth a single rule, a single axiom for grasping the objective world
it would be this: “No paradoxes.” In its weaker form, it is
simply a requirement for consistency. A well-curated library of
objective data bases is expected to self-consistent. The computations
that one can perform should be incapable of generating paradoxes.
Where conflicts seem to occur, they can be resolved by showing that
they occur in mutually exclusive environments. This would be in the
spirit of Niels Bohr’s approach for handling phenomena that seems
paradoxical yet is instead complementary.

In a subjective murky world in which
one’s information leaks from dubious and duplicitous sources,
self-contradicting and self-defeating statements arise. Irrational
behavior can be explained. Maybe not to someone doing it, but to
onlookers, its presence can be reasoned out; that is what the “No
paradoxes” axiom proclaims. It’s a natural meta-rule for
mathematics and science. If the axiom holds up over fuzzy and
subjective emotions, then anything can be reasoned out. This is not
the same as saying there is a reason for everything.

I think Tegmark is conflating the signifier with that which it signifies - that the mathematics which describe the observable phenomenon are in some mystical sense the root of the observable phenomenon. "Ceci n'est pas un pipe" - words on a picture of a pipe - but the picture only depicts the pipe, you wouldn't want to try to smoke it.

As for the electron spin, the observation came first, and then the mathematics followed:

"When the day came I had to tell Uhlenbeck about the Pauli principle - of course using my own quantum numbers - then he said to me: "But don't you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates". Now, I can also exactly tell you the difference between Uhlenbeck and me as physicists. In those days, all through the summer when I told Uhlenbeck about Landé and Heisenberg, for instance, or about Paschen, then he asked: "Who is that?" He had never heard of them, strange. And when he said: "That means a fourth degree of freedom", then I asked him: "What is a degree of freedom?" In any case, when he made his remark, it was luck that I knew all these things about the spectra, and I then said: "That fits precisely in our hydrogen scheme which we wrote about four weeks ago. And if one now allows the electron to be magnetic with the appropriate magnetic moment, then one can understand all those complicated Zeeman-effects. They come out naturally, as well as the Landé formulae and everything, it works beautifully".

The observation came first, the realization that a fourth degree of freedom, electron spin, could be used to model the observation, came second. The mathematics was fitted up to the observation, not the other way around.

Oh, and as to Tegmark's need for an "additional principle in his arxiv paper (http://arxiv.org/abs/1401.1219) of 6th January 2014, might I suggest the idea that the brain is a transceiver of consciousness, not a producer of it, and that consciousness might better be regarded as a field rather than as a form of matter? The article seems to be a reductio proof of that.

Thanks for raising this interesting point of possible conflation between the signifier and the signified. You’re quite right we humans invent the *language* of mathematics (the symbols, our human names for the symbols, etc.), but it’s important not to conflate this language with the *structure* of mathematics - it's only the latter which you might argue corresponds to the physical world.
For example, any civilization interested in Platonic solids would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist. The same applies to the mathematical structures that are popular in modern physics, from 3+1-dimensional pseudo-Riemannian manifolds to Hilbert spaces. Once we’ve classified all irreducible representations of the Poincare group, we’re not free to invent new ones!
/Max

OK, so you're talking about category theory, right?
"Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.

I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context." http://johncarlosbaez.wordpress.com/2013/05/23/category-theory-for-scien...

BTW, here are a couple of things on arxiv.org you might want to look at - if you're not familiar with them already:http://arxiv.org/abs/1302.6946 "There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity." It's by David Spivak, he's an MIT prof, btw.

Prof. P. writes, “Mathematical Platonism is the idea that mathematical structures are real in a mind-independent fashion.” What is an idea? What are mathematical structures? What is real? What is mind-independent? My guess is that superhuman intelligence would be required to satisfactorily answer the preceding 4 questions at the human level. Read the books of Oliver Sacks if you don’t believe me.
Prof. T. writes, “Without infinity, there are, as you say, no Gödel issues in our physics.” Without a physical infinity, there would seem to be no Gödel issues in the non-psychological mechanisms of physics, but psychological mechanisms might be an inherent aspect of physics.
I request feedback (abusive feedback OK) on the following:
“La semplicità è la massima raffinatezza.” (Simplicity is the ultimate sophistication.) — Leonardo da Vincihttp://artezza.altervista.org/la-citazione-del-giorno-17/
“If you can’t explain it simply, you don’t understand it well enough.” — Einstein
What is quantum gravity all about? Photons and gluons can’t escape from the universe in which they are located. Gravitons travel at the speed of light on average. A statistically significant few gravitons travel slower than the speed of light. These slow gravitons cause the Fernández-Rañada-Milgrom effect. A statistically few gravitons travel faster than the speed of light and escape from the boundary of the multiverse into the interior of the multiverse. These fast gravitons cause the nonzero cosmological constant and the inflaton field. Electromagnetic radiation from the inflaton field shows up as the space roar. If the fast gravitons never escaped from the universe in which they are located, then the slow gravitons and the fast gravitons would average out, yielding Einstein’s field equations with cosmological constant = zero and dark-matter-compensation-constant = zero. Is the preceding scenario a mélange of nonsense?

I must admit that I am very much in the camp of a realist, (as was Einstein) I particularly like your basic description:"To put it very briefly, a realist is someone who thinks that scientific theories aim at describing the world as it is (of course, within the limits of human epistemic access to reality), while an anti-realist is someone who takes scientific theories to aim at empirical adequacy, not truth." [emphasis my own]

I just might have to pick up the book that you referenced, it looks like it may be a good entertaining read.

Hard scientific evidence is hard scientific evidence! You make an electron out of light in pair production. You annihilate it and get light again. Et cetera. There is no calculated prediction to this hard scientific evidence. You must be a mathematician, not a physicist. Which explains your scant physics knowledge, and your unwillingness to criticize a universe made of mathematics. Name and rank please! What's the problem?

...This is not an ad-hominem attack. It is an attack on your logic (or lack there-of in this case)...

There's no lack of logic in pointing out pair production and annihilation and Einstein-de Haas and magnetic moment and electron diffraction and refraction and atomic-orbital standing waves then pointing out the wave nature of matter. The lack of logic is yours, in ignoring all this evidence, in confusing mathematics with evidence, and repeatedly asserting that the electron is a point particle.

Your belief that the invariant mass of a system decreasing means the invariant mass of the particles decreased ... that is NOT "robust mainstream physics". That is a misunderstanding of physics, and is very much "your idea".

It is robust mainstream physics. You just don't know enough of it.

You can try to define such a thing, but in general in GR...

It not a question of defining such a thing. It's simple matter of doing work on the brick when you lift it. You expend energy, and conservation of momentum means the brick gets the lion's share of that energy. Lift the brick at 11km/s and it ends up taking it away.

...until you understand the basics of GR first...

I understand GR. I'm the one who quotes Einstein. You don't. What you understand is some ersatz GR that is a corruption of Einstein's theory.

Conservation of energy in GR does NOT work like you apparently think it does.

Oh yes it does. There is no way to get rid of energy or create it ex nihilo.

Let's try a simple counter example, consider the free falling coordinate system in which the particle is at rest. It's kinetic energy is zero for the entire trip, and there is no force on it. It's invariant mass is constant. Do you still deny this?

Yes I do. A coordinate system is an abstract thing. It is an artefact of measurement. It does not exist. But you exist, and you are falling alongside the particle, whereupon you measure its mass using some other particle. Hence you measure the same mass. You also measure zero kinetic energy. However when you impact the ground at 11km/s your illusions concerning zero kinetic energy are shattered, and in your final moment you appreciate that the energy that destroys you comes from you.

If so, actually work out the math for the prediction and post it in a blog here on science20.

No! Drop the 1kg brick into a black hole and the black hole mass increased by 1kg. At all times the "rest" mass plus the kinetic-energy mass-equivalence adds up to 1kg. It's that simple. Your attempt to hide behind a mathematical smokescreen will not get you off this hook.

Again, I ask: What could make you consider you are wrong if you are unwilling to actually calculate the predictions of GR?

An honest answer from you. You fall. There is no external force acting upon you. No energy is being added to you. Then BANG, you hit the ground in a spectacular fashion. Now, where did all that kinetic energy come from?

You must be a mathematician, not a physicist. Which explains your scant physics knowledge, and your unwillingness to criticize a universe made of mathematics.

I agree with Massimo's and Scott Aaronson's criticisms of the "Mathematical Universe" idea. You however are arguing that you don't need to use math at all. In science, math is incredibly useful for precisely describing our hypothesis, from which we can calculate the theory's predictions and compare it to experiment. You instead are just making statements about what physics predicts that are incorrect, both in that they do not actually derive from the theories you claim they do as well as some of your ideas contradicting experiment. Regardless of the "Mathematical Universe" idea, math is a necessary tool in science. Massimo and others, in their criticisms were not arguing that physicists should stop using math. I don't think you are either, but you sure do complain like it when you repeatedly refuse to show the math backing up your claims.

To claim GR predicts such-and-such is a mathematical claim. That is the beauty of specifying theories precisely in the language of math. We can check such a claim using just math. To then check the theory with nature, requires experiment. Before you even do the check with experiment, you have to get the math correct to calculate the prediction. This is the step in which you are failing. You are making claims that are mathematically incorrect.

Hard scientific evidence is hard scientific evidence! You make an electron out of light in pair production. You annihilate it and get light again. Et cetera. There is no calculated prediction to this hard scientific evidence.

You are listing observations. There is nothing wrong with that. The problem is you keep claiming these observations support your hypothesis (an electron is a confined photon). That is NOT "hard scientific evidence". You just made a leap with no logic connecting it. Specify a theory precisely in math, calculate a prediction of this theory, and compare it to experiment -- THAT would be scientific evidence. Currently quantum electrodynamics matches all experimental tests -- THAT is scientific evidence. But you reject this, not based on experiment, but on "a priori" metaphysical beliefs that you have. That is not science. I am not rejecting your "hard scientific evidence" for your ideas, for there is none; I am rejecting your incredibly flawed logic.

Everything else in your post is you spouting your misunderstanding of physics again. You say you will consider you are wrong if I give an honest answer to one of your questions, so I will do so and hope that you finally respond to the multiple requests you've had now to write a blog showing the math backing your claim that GR predicts the invariant mass of a free falling point particle to change along its path.

An honest answer from you. You fall. There is no external force acting upon you. No energy is being added to you. Then BANG, you hit the ground in a spectacular fashion. Now, where did all that kinetic energy come from?

Here is an honest answer: The question presupposes an answer and is flawed. It is equivalent to asking: "In Newtonian mechanics, two particles A and B are moving inertially. There is no external force acting on either particle. They collide. Now, where did all that kinetic energy come from?"

Using an inertial coordinate system in which A was at rest, it would be perfectly correct to say A was at rest with no kinetic energy until the collision, and particle B had all the kinetic energy until the collision. However, using an inertial coordinate system in which B was at rest, it would be perfectly correct to say B was at rest with no kinetic energy until the collision, and particle A had all the kinetic energy until the collision. Using yet a different coordinate system, both A and B have kinetic energy. So, where did all that kinetic energy come from? The question itself is flawed, as it presupposes there is a definitive answer, while there is not. Kinetic energy is a coordinate system dependent concept.

The invariant mass of a point particle is a constant while it free-falls in GR. Using a local inertial frame in which the free-falling particle is at rest (this does not require me to physically fall with the particle to use such a coordinate system to describe the events as you somehow seem to imagine, we are free to choose the coordinate system to describe the events), using this local inertial frame the energy is constant and the momentum is constant and (mc^2)^2 = sum over u, v (P^u P^v g_uv) = E^2 - (pc)^2 is a constant while falling (where P^u is the four momentum, g_uv is the metric). Using instead the Schwarzschild coordinate system, the spatial components of the four-velocity of the test particle are no longer zero, yet the mass (mc^2)^2 = sum over u, v (P^u P^v g_uv) is still the same constant invariant, but -- and maybe this is where your math error is -- in Schwarzschild coordinates, the metric isn't the simple inertial coordinate one having on the diagonal 1,-1,-1,-1. The math is more complicated, but again the invariant mass is a constant.

If you disagree, and still claim that GR predicts that the invariant mass of a test particle changes as it free-falls, then please write a blog showing your math backing your claim. Confine your pet theories to your blog, and people can respond there.

John, so do you still disagree with the predictions of GR? If my explanation above helped us gain some common ground, please let me know. If not, I'm hoping you are working on your blog to show the math backing up your claims. It will be an interesting read.

I don't disagree with the predictions of GR. This conversation about the mass deficit started after I referred to mass in general relativity. See this:

"In special relativity, the invariant mass of a single particle is always Lorentz invariant. Can the same thing be said for the mass of a system of particles in general relativity? Surprisingly, the answer is no..."

Real quick, your quote regarding the mass of a system of particles doesn't back up your claims about the invariant mass of a particle changing as it free falls. Furthermore, the issue referred to in that quote is related to the extent to which we can say energy is conserved in GR when we have time dependent sources. Do you want to argue that energy is not conserved when a particle free falls in a Schwarzschild background? If not, then stop trying to use that quote as if it backs your claims. Also, your idea that the kinetic energy of a falling mass somehow comes at the expense of its invariant mass is not predicted by GR and is completely unrelated to the issue that quote is referring to. Let's first focus on getting agreement of the predictions for a single particle moving in a simple solution of GR.

"I don't disagree with the predictions of GR."

Then you should agree that the invariant mass of a point particle moving in a Schwarzschild background will be an invariant constant. Do you indeed agree with that now? The mathematical predictions of a mathematical theory are not up for debate. Your ideas about kinetic energy coming from a reduction of invariant mass are wrong.

If you disagree, write a blog calculating the predictions of GR to back up your claims on what GR predicts. You are making a mathematical claim. Show the math. In doing so, you will hopefully even find your error yourself.

No I do not agree with that now. And no I will not write a blog. And no I am not saying you don't need to use math at all. Mathematics is a vital tool for physics. But it is not what physics is. Conservation of energy is physics. And the kinetic energy dissipated as radiation, and the mass deficit, and a radiating body loses mass. I've read the rest of your post, and it's assertions and aspersions and Humpty Dumpty physics that totally fails to address the issue. The system lost mass, and it could be a system of two particles only. So those particles lost mass. Your "honest answer" refers to coordinate systems which do not in themselves exist, and you refer to inertial coordinate systems which blinds you to reality wherein you end up evading the question. The metric is an abstract artefact describing what you measure. Your measurements leave you thinking the mass is unchanged, but it is changed. There is a deficit, you have not accounted for it, and you know this. And so you know that your answer is not an honest answer. And since you continue to talk of "my ideas" and you still won't give your name, it's clear that you aren't honest either.

You say you don't disagree with the predictions of GR, but you refuse to actually calculate the predictions yourself, and dismiss anyone else showing how you can calculate it because they use a "metric" or "coordinate systems". In short you ARE disagreeing with the predictions of GR.

As multiple people have tried to explain now, the predictions of a mathematical theory are not up for debate. You have many bizarre ideas, and the reason so many people are zooming in on this particular GR claim, is that your speculations on electrons being made of photons and other ideas are so vague that it is hard to pin you down to specifics to get you to understand how wrong the idea is. The GR claim however is already a mathematical claim. You are claiming a mathematical theory predicts the invariant mass of a particle decreases as it free-falls in order for its kinetic energy to increase. That is wrong. Anyone that can calculate the predictions of GR can do the math and check that, sure enough, you are wrong. Pure and simple. You won't trust anyone else explaining the math, and you refuse to do the math yourself. Seriously, what could possibly get you to understand that the predictions of GR do not match your claims if you refuse to discuss how to calculate the prediction?

Above, it almost appears that you don't even agree on the mathematical definition of the invariant mass. The mathematical definition is:

Where m is the invariant mass, is the momentum four-vector, and is the metric.Do you disagree with the very definition of invariant mass?

In an inertial coordinate system this simplifies to

which may be the form you are more familiar with, but it is a special case.

"A generic prediction of chaotic inflation is an INFINITE ergodic universe, which, being INFINITE, must contain Hubble volumes realizing all initial conditions. Accordingly, an INFINITE universe will contain an INFINITE number of Hubble volumes, all having the same physical laws and physical constants. In regard to configurations such as the distribution of matter, almost all will differ from our Hubble volume. However, because there are INFINITEly many, far beyond the cosmological horizon, there will eventually be Hubble volumes with similar, and even identical, configurations. Tegmark estimates that an identical volume to ours should be about 2^10^118 meters away from us. [4] Given INFINITE space, there would, in fact, be an INFINITE number of Hubble volumes identical to ours in the universe. [5] This follows directly from the cosmological principle, wherein it is assumed our Hubble volume is not special or unique".

Good point.
This is what you get when someone starts revising their ideas on the fly to counter criticisms.

Max needs to clarify what he means now given that he rejects basically all the math concepts used in physics models from existing in his "Mathematical Universe". So why is he so sure math concepts can "be" the universe around us if it is not even clear if his limited allowed math structures can even model the universe around us? I honestly don't think he could give a mathematical model for the universe that could replace the theories of modern physics which use real numbers, continuous lie groups, etc. Until he is proposing something concrete, what he's proposing is not a scientific hypothesis.

When people invoke the notion of infinity, the concept of things being unimaginably large comes to mind. In physics, when a theory predicts a measureable quantity to go to infinity, I've yet to meet someone who believes the theory's predictions in that case are still within its region of applicability.

However, infinite in the sense of something unimaginably small, such as all the little steps between 0 and 1 ... that has some very real consequences on theories. Some continuous symmetries have been tested below even the planck scale. So what in the world are you actually saying about infinities?

Are you saying real numbers are not allowed in your "Mathematical Universe"? Not even rational numbers? No continuous symmetries? Not even integers?

If you force use to use a finite set of integers, and only discrete symmetries, I have to say ... your "Mathematical Universe" sounds very VERY unlike the world around us.

Sure, I'll do my best.
Hopefully this will give some intuitive understanding, but as with all such introductions realize not to take some analogies too literally.

Consider a continuous homogeneous elastic medium. In this idealized object, we could calculate the speed of disturbances in the material ... the speed of sound. With an ideal elastic material, this would just be a constant, independent of frequency. In a continuous medium, we can imagine a wave of any spatial frequency rippling through it.

If however, we consider an elastic medium built of discrete pieces, say a bunch of little spheres connected with ideal springs, now certain frequencies don't even make "sense". I don't know your background, but even intro level engineering courses usually run into Fourier series, and the Nyquist limit. Basically, waves with wavelength well below the spacing of the spheres can't occur in this discrete material.

Note that in lower energy, the acoustic modes look almost linear in energy. This is the low energy "E = const freq" relation.

A similar thing happens for light. For light we expect "E = h f" for a photon, but this would no longer be exact if space was not continuous; there would be a dispersion relation. Fairly general in such theories, higher energy light will travel slower. Over vast differences in outer space, this can _REALLY_ add up. It is these vast distances, and high energy pulses of electromagnetic energy from astrophysical sources, that allows limits to have been measured to the planck scale.

Yes that does help. Whats your position on Loop quantum gravity then, isn't that meant to be discrete and predict slower light for very high freqs? Have such experiments over vast distances been a test for it, I seem to remember an article that claimed high freq gamma rays had been measured going slower with billion year old supernova or something but I can't find it now.

LQG can be a touchy subject. It can get some physicists riled up. I have not studied the math behind it well enough to have a strong opinion, but I'll give some info where I can.

predict slower light for very high freqs

People interested in LQG initially didn't even agree on this prediction. Lee Smolin pushed this for a bit, even suggesting a "Doubly-Special" relativity that many people have pointed out doesn't appear to be very well behaved mathematically. Rovelli seems to not take a strong stand, sometimes referring to the possibility of calculating a dispersion and other times seeming to suggest something along the lines of the spectrum of the geometry operators are quantized, but the spacetime itself is continuous.

From an outsider's perspective the researchers in this field seem to often not even be talking about the same thing, and then "reformulate" the approach every once in awhile. For example the Barret-Crane model was shown to not have the correct long distance limit, the model was rewritten, and they started over.

isn't that meant to be discrete

The simplified layman's description just straight out claims "space atoms". More accurately though it is the spectrum of the operators that is quantized. While the area operator is quantized, it doesn't necessarily mean space is a bunch of "space atoms", and they definitely don't add together the way you'd imagine some classical "space chunks". A common point made along these lines is that angular momentum quantization does not mean angular rotations are quantized. Here's a paper by Rovelli discussing some of this:http://arxiv.org/pdf/gr-qc/0205108v1.pdf

The original "loops" in loop quantum gravity are actually integrals along a loop in spacetime (wilson loops), and it is these loops that are quantized. So in my mental picture, it doesn't mean anything to talk about patches smaller than the minimum quantized area, but there was nothing special about what coordinate system we initially considered drawing these loops in. So there is no preferred frame per say. Please understand that my understanding of these topics is not deep, and since to me, even researchers in the field seem to give conflicting sounding responses, I may very well be cranky here. Our maybe the field hasn't come to concensus. If someone wants to give a more definitive response that would be nice (hopefully point to an article that makes it clearer or more precise, some math even).

Have such experiments over vast distances been a test for it, I seem to remember an article that claimed high freq gamma rays had been measured going slower with billion year old supernova or something but I can't find it now.

I remember something with a gamma ray burst, but think it was more a 'maybe something is here let's follow up' kind of thing and not a definitive measurement. Many people pounced on it, including Smolin saying this was a test of Loop quantum gravity's photon dispersion. Others in LQG balked at this though. Later when a lot more data from gamma ray bursts were taken (I think with a different satellite more suited for the measurements) no dispersion effect was seen.

Massimo,You let Max off the hook a bit by saying "The question of whether there really are infinities in physical systems is open, so surely Max is entitled to his skepticism."

But our current descriptions of physics use real numbers, and the notion of continuous, all the time. For instance, to expand on LFB's comments above, Max is basically saying the continuous U(1) symmetry in Quantum Electrodynamics is only approximate. To back this up he would need to provide a new model of electrodynamics. This is not just a philosophical issue then of how we consider the role of math in science. This would mean a real change to the standard model. Even string theory could not be used.

His idea when viewed this way sounds very testable. In fact, I think he'd be hard pressed to come up with a mathematical theory that can match all current experimental data, and yet only use discrete and finite fields, symmetries, etc. So difficult in fact, that this almost sounds impossible to me. This is in great conflict with current theories of physics.

Would you care to address some of these views in another article? It seems to open up a whole new line of discussion on Max's "Mathematical Universe" idea ... it takes it from what appeared to be so vague as to be untestable, to being (ironically) violently at odds with current foundations of mathematical physics.

Since his idea rules out the standard model of physics, unless he actually submits an alternative to the standard model of physics, I don't consider this even a theory. Saying all of current physics is wrong, and not supplying an alternative, well... I don't consider that science.

Dear Massimo,
I enjoyed your article, and the follows up by Max and some readers here really add to it (I guess that's why they call it Science 2.0 !). However my understanding of Godel's incompleteness theorem is too weak to piece all this together.

What exactly does Godel's theorem have to do with infinities and (as suggested above) real numbers, or even all the integers?
If the connection is mathematically true, then this really does seem to make his theory not only testable, but actually on the verge of being experimentally ruled out. For all I know, there's already a "no-go" theorem saying a finite field ruins the action principle, and Hamiltonian dynamics, and so on. For someone claiming the world is Mathematical structures, Max sure seems to be capriciously eliminating all the structures that we've found can be used to model the world. So unless he can actually supply at least ONE such structure to describe the physics around us, his theory appears stillborn.

How is this idea NOT dead?
I'd absolutely love a follow up article going more into the relation with physics, philosophy, and Godel's theorem.

In your writing, your curiosity and enthusiasm comes across as wonderful and sincere. Such explorations with math and modelling can be wonderful ways to exercise the mind and have fun learning. This should be encouraged. I do encourage you to continue exploring and continue learning.

The issues raised by your writing however are not whether "it is correct or not". What we have here is a "my theory" problem. As currently laid out, someone could rework through your simple algebra, or rerun a simulation, but it is too vague to allow anyone else but you to predict anything with it. This is a common pseudo-science trap. For instance you make claims like

In This essay I shall derive the laws of nature from a simple mathematical system. The system is derived from the postulate that reality is nothing but a mathematical structure which leads to a simple system that can be
simulated to generate many results. The postulate lead to assume particles as made of ...

and

...ultimately producing the correct lagrangian automatically...

These claims are not correct. You are not actually deriving things here. You can't actually derive your statements from "the postulate that reality is nothing but a mathematical structure". As you go, you add new ideas, new relations, new definitions, and new interpretations.

What you actually have presented is just your thought process as you explored starting from some idea. This may be interesting, but it is no more a mathematical derivation than word association is a derivation.

So, as it stands, no one can assess your "theory", because it is not a well developed hypothesis yet. If you want to give the background of how you came about some idea to give some background and make it more entertaining, that is fine (although usually distracting and not recommended if you want to clean this up to try to publish in a journal). That background however is not important. Ultimately what matters is your idea should be precisely stated so that anyone can unambiguously work out the predictions of the theory. Then, from the hypothesis, derive (directly, and mathematically; no going and adding things to your hypothesis at this point) the predictions, and compare those predictions to either experiment directly or (even better in this case for a theory paper) compare how these predictions align (or not) with current well tested modern physics theories.

If you rewrite your essay to fix this "my theory" symptom and problem, other people will be able to assess it. For instance, if I wanted to calculate all the energy levels for the hydrogen atom, in particular the lamb shift, how would I do so? How are units even introduced here? You mention the mass of the electron, how do you know it is an electron? How do I calculate the mass of the muon, or tau? And so on and so on. Currently due to the "my theory" problem, no one can "derive" any predictions from your theory but you.

Until your hypothesis is stated clearly enough to be a scientific hypothesis, it is meaningless to ask whether it is 'correct' scientifically. So please don't fall in that trap. You seem to be enjoying yourself, and are creative, so keep exploring, just be mindful of the work that needs to be done before anyone can really assess your work. And equally important, I hope you understand now why this extra work is necessary (it is NOT an aesthetic issue, but a real issue of logic and science). I hope this long explanation helps and fits in your ".01% for other possibilities" of expectations.

Thank you very much for your review of my idea (I hope that word works better). I accept your assessment as fair enough since the presentation of my "idea" is what it is at this time. I will reply in detail tomorrow.

There are no physical infinities. And
as Nietzsche noted, no perfectly straight lines, no perfect circles,
not even absolute magnitudes. That in itself, does not imply that the
standard MODEL is in conflict with mathematics. It simply means that theoretical physics takes in its scope and bag of tools, more than just physical realities. The only question
here is whether physical universes fit completely within
mathematics, as in being mere subsets, or whether, as I queried
above ... Do physical worlds extend beyond mathematics or at least
what is computable?

Is that not why Massimo does not fully include himself in Max's Mathematical Universe?

These questions are the same as
asking whether there are limits to artificial intelligence even-just-simulating what goes on in the real worlds of human events and
affairs. I will add a bit more … before this thread dies. I wonder
what Mass and Max think. Sorry to seem them go … without anymore
follow up.

I looked back and tried to see again
just why Massimo is “not convinced”. There is the obvious
difficulty on whether Max's thesis is testable. My reasons for not
being convinced are rooted in whether our human affairs are
fundamentally computable, rational, reasonable or structured in every
form and aspect. There are structures and regularities aplenty, and
yet, not everything is proven to be structured.

It is a difficult challenge to convince
a Science20 audience that quantitative reasoning cannot cover all
that is knowable about our physical universe. Over in the humanities,
I doubt if anyone is persuaded that human affairs are completely and
totally mathematically structured. That is not proof, however, that
there are experiences that mathematics cannot touch. Mind you, I
myself, majored in mathematical physics from within a math
department, so I have already kissed the mathematical blarney stone.

So here we are in a science site were
everyone believes our worlds are fundamentally logical and rational,
and yet, who are we kidding if we are to survive?

If you look at what published
scientists actually do, you'll quickly see in their rationalizing
that their favorite tool is a footnote or a hyperlink. This
cross-indexing is a mathematical structure. It is also a bit of
cheat. It is not profound and easily abused. It is only a step away
from the way people freely associate in every day conversations. The
purported structure itself can be quite annoying. In Massimo's essay
at the top, he has lots of hyperlinks and yet, none of them get to
the meat as to why he is not convinced that our Universe is
fundamentally mathematical.

If Max Tegmark's use of the word
mathematical is so broad and inclusive that it swallows up
everything, even random cross-indexing and Alice in Wonderland
gibberish, then it becomes a rather empty term.

Surely the logician and mathematician
Lewis Caroll was aware of there being fertile worlds over and beyond
mathematics when he fell down a rabbit hole and created Alice in
Wonderland.

Vast amounts of the world as we know
it—or could know it—is objective and computable … in principle.
This is not a radical hypothesis. It is clashes however with the
clouds of ambiguous and spontaneous notions that rule us. The masses
are not as informed as they could be from rigorous and impartial
calculations on hard and curated data. It's a false hope, however,
that they every will be or could have been.